Handout: twbh.html, twbh.pdf, twbh.png. Sources: twbh.zip, pensieve.
Abstract. I will be talking about an invariant ζ. For the first 15 minutes I will be talking about its target space, algebra (trees and wheels, or free Lie algebras and cyclic words). For the next 15 minutes I will talk about its domain space, topology (knotted balloons and hoops in 4-space). And in the remaining time I will tell you why I care, though with little detail: It is the universal solution to a topological problem and it has many siblings (who talk to each other). It is explicitly computable. Its target space is in itself a space of "universal formulas in Lie algebras" (that's "the miracle"). It seems to be a complete(?) evaluation a certain gauge theory. It is related to a deep conjecture in Lie theory proven by Alekseev and Meinrenken. It has even-better-computable specializations, including one which is an "ultimate Alexander invariant". And plenty of work remains to be done.
Talk video. 
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There's also a paper in progress.
